Surfaces and Gradients

u

-3

v

-0.14

w

1.405

Visualize how gradients relate to surfaces.

The points

(x,y,z)

satisfying

w=f(x,y,z)=u

2

x

+v

2

y

+

2

z

for a particular value of

w

form the surface shown. As

w

varies, the surface deforms along the normals defined by the vector field

∇f=

∂f

∂x



i

+

∂f

∂y



j

+

∂f

∂z



k

, the gradient of

f

, represented by the arrows.

Due to the complex nature of this Demonstration, there may be a delay between changing the variables and updating the graph.